Minimally Intersective Polynomials with Arbitrarily Many Quadratic Factors

TL;DR

This paper constructs infinitely many polynomials with multiple quadratic factors that have roots modulo every positive integer but no rational roots, and shows the set of such polynomials is dense in natural numbers.

Contribution

It provides an explicit construction method for minimally intersective polynomials with many quadratic factors and analyzes their density among natural numbers.

Findings

Existence of infinitely many such polynomials for each n ≥ 4.

Explicit construction process for these polynomials.

The set of suitable a_n has positive asymptotic density.

Abstract

Given a natural number $n \geq 4$ we show that there exists infinitely many polynomials $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ such that (i) $f_{n}(x)$ has a root modulo every positive integer, (ii) $f_{n}(x)$ has no rational roots, and (iii) every proper divisor of $f_{n}(x)$ fails to have root modulo some positive integer. We exhibit a process to explicitly construct such $f_{n}$ and this process demonstrates that the set of natural numbers $a_{n}$, such that the polynomial $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ satisfies the properties (i), (ii) and (iii), is of positive asymptotic density in $\mathbb{N}$.